?(a) Find the partial sum \(s_{10}\) of the series \(\sum_{n-1}^{\infty} 1 / n^{4}\)

Draw a picture to show that \(\sum_{n=2}^{\infty} \frac{1}{n^{1.5}}<\int_{1}^{\infty} \frac{1}{x^{1.5}} \ d x\) What can you conclude about the series? Equation Transcription: Text Transcription: Sum over n=2 ^infty 1/n^1.5 < integral over 1 ^infty 1/x^1.5 dx

Suppose \(f\) is a continuous positive decreasing function for \(x \geqslant 1\) and \(a_{n}=f(n)\). By drawing a picture, rank the following three quantities in increasing order: \(\int_{1}^{6} f(x) d x \quad \sum_{i=1}^{5} a_{i} \quad \sum_{i=2}^{6} a_{i}\) Equation Transcription: ? Text Transcription: f X geqslant 1 a_n=f(n) Integral over 1 ^6 f(x)dx sum over i=1 5^ a_i sum over i=2 ^6 a_i

Use the Integral Test to determine whether the series is convergent or divergent. \(\sum_{n=1}^{\infty} n^{-3}\) Equation Transcription: Text Transcription: Sum over n=1 ^infty n^-3

Use the Integral Test to determine whether the series is convergent or divergent. \(\sum_{n=1}^{\infty} n^{-0.3}\) Equation Transcription: Text Transcription: Sum over n=1 ^infty n^-0.3

Use the Integral Test to determine whether the series is convergent or divergent. \(\sum_{n=1}^{\infty} \frac{2}{5 n-1}\) Equation Transcription: Text Transcription: Sum over n=1 ^infty 2/5n-1

Use the Integral Test to determine whether the series is convergent or divergent. \(\sum_{n=1}^{\infty} \frac{1}{(3 n-1)^{4}}\) Equation Transcription: Text Transcription: Sum over n=1 ^infty 1/(3n-1)^4

Use the Integral Test to determine whether the series is convergent or divergent. \(\sum_{n=2}^{\infty} \frac{n^{2}}{n^{3}+1}\) Equation Transcription: Text Transcription: Sum over n=2 ^infty n^2/n^3+1

Use the Integral Test to determine whether the series is convergent or divergent. \(\sum_{n=1}^{\infty} n^{2} e^{-n^{3}}\) Equation Transcription: Text Transcription: Sum over n=1 ^infty n^2e^-n^3

Use the Integral Test to determine whether the series is convergent or divergent. \(\sum_{n=2}^{\infty} \frac{1}{n(\ln n)^{3}}\) Equation Transcription: Text Transcription: Sum over n=2 ^infty 1/n(ln n)^3

Use the Integral Test to determine whether the series is convergent or divergent. \(\sum_{n=1}^{\infty} \frac{\tan ^{-1} n}{1+n^{2}}\) Equation Transcription: Text Transcription: Sum over n=1^infty tan^-1 n/1+n^2

Determine whether the series is convergent or divergent. \(\sum_{n=1}^{\infty} \frac{1}{n^{\sqrt{2}}}\) Equation Transcription: Text Transcription: Sum over n=1 ^infty 1/n^sqrt 2

Determine whether the series is convergent or divergent. \(\sum_{n=3}^{\infty} n^{-0.9999}\) Equation Transcription: Text Transcription: Sum over n=3 ^infty n^-0.9999

Determine whether the series is convergent or divergent. \(1+\frac{1}{8}+\frac{1}{27}+\frac{1}{64}+\frac{1}{125}+\cdots\) Equation Transcription: Text Transcription: 1+1/8+1/27+1/64+1/125+cdots

Determine whether the series is convergent or divergent. \(\frac{1}{5}+\frac{1}{7}+\frac{1}{9}+\frac{1}{11}+\frac{1}{13}+\cdots\) Equation Transcription: Text Transcription: 1/5+1/7+1/9+1/11+1/13+cdots

Determine whether the series is convergent or divergent. \(\frac{1}{3}+\frac{1}{7}+\frac{1}{11}+\frac{1}{15}+\frac{1}{19}+\cdots\) Equation Transcription: Text Transcription: 1/3+1/7+1/11+1/15+1/19+cdots

Determine whether the series is convergent or divergent. \(1+\frac{1}{2 \sqrt{2}}+\frac{1}{3 \sqrt{3}}+\frac{1}{4 \sqrt{4}}+\frac{1}{5 \sqrt{5}}+\cdots\) Equation Transcription: Text Transcription: 1+1/sqrt 2+1/3 sqrt 3+1/4 sqrt 4+1/5 sqrt 5+cdots

Determine whether the series is convergent or divergent. \(\sum_{n=1}^{\infty} \frac{\sqrt{n}+4}{n^{2}}\) Equation Transcription: Text Transcription: Sum over n=1 ^infty sqrt n+4/n^2

Determine whether the series is convergent or divergent. \(\sum_{n=1}^{\infty} \frac{\sqrt{n}}{1+n^{3 / 2}}\) Equation Transcription: Text Transcription: Sum over n=1 ^infty sqrt n/1+n^3/2

Determine whether the series is convergent or divergent. \(\sum_{n=1}^{\infty} \frac{1}{n^{2}+4}\) Equation Transcription: Text Transcription: Sum over n=1 ^infty 1/n^2+4

Determine whether the series is convergent or divergent. \(\sum_{n=1}^{\infty} \frac{1}{n^{2}+2 n+2}\) Equation Transcription: Text Transcription: Sum over n=1 ^infty 1/n^2+2n+2

Determine whether the series is convergent or divergent. \(\sum_{n=1}^{\infty} \frac{n^{3}}{n^{4}+4}\) Equation Transcription: Text Transcription: Sum over n=1 ^infty n^3/n^4+4

Determine whether the series is convergent or divergent. \(\sum_{n=3}^{\infty} \frac{3 n-4}{n^{2}-2 n}\) Equation Transcription: Text Transcription: Sum over n=3 ^infty 3n-4/n^2-2n

Determine whether the series is convergent or divergent. \(\sum_{n=2}^{\infty} \frac{1}{n \ln n}\) Equation Transcription: Text Transcription: Sum over n=2 ^infty 1/n ln n

Determine whether the series is convergent or divergent. \(\sum_{n=2}^{\infty} \frac{\ln n}{n^{2}}\) Equation Transcription: Text Transcription: Sum over n=2 ^infty ln n/n^2

Determine whether the series is convergent or divergent. \(\sum_{k=1}^{\infty} k e^{-k}\) Equation Transcription: Text Transcription: Sum over k=1 ^infty ke^-k

Determine whether the series is convergent or divergent. \(\sum_{k=1}^{\infty} k e^{-k^{2}}\) Equation Transcription: Text Transcription: Sum over k=1 ^infty ke^-k^2

Determine whether the series is convergent or divergent. \(\sum_{n=1}^{\infty} \frac{1}{n^{2}+n^{3}}\) Equation Transcription: Text Transcription: Sum over n=1 ^infty 1/n^2+n^3

Determine whether the series is convergent or divergent. \(\sum_{n=1}^{\infty} \frac{n}{n^{4}+1}\) Equation Transcription: Text Transcription: Sum over n=1 ^infty n/n^4+1

Explain why the Integral Test can’t be used to determine whether the series is convergent. \(\sum_{n=1}^{\infty} \frac{\cos \pi n}{\sqrt{n}}\) Equation Transcription: Text Transcription: Sum over n=1 ^infty cos pi n/sqrt n

Explain why the Integral Test can’t be used to determine whether the series is convergent. \(\sum_{n=1}^{\infty} \frac{\cos ^{2} n}{1+n^{2}}\) Equation Transcription: Text Transcription: Sum over n=1^infty cos^2n/1+n^2

Find the values of \(p\) for which the series is convergent. \(\sum_{n=2}^{\infty} \frac{1}{n(\ln n)^{p}}\) Equation Transcription: Text Transcription: p Sum over n=2 ^infty 1/n(ln n)^p

Find the values of \(p\) for which the series is convergent. \(\sum_{n=3}^{\infty} \frac{1}{n \ln n[\ln (\ln n)]^{p}}\) Equation Transcription: Text Transcription: p Sum over n=3 ^infty 1/n ln n [ln(ln n)]^p

Find the values of \(p\) for which the series is convergent. \(\sum_{n=1}^{\infty} n\left(1+n^{2}\right)^{p}\) Equation Transcription: Text Transcription: p Sum over n=1 ^infty n(1+n^2)^p

Find the values of \(p\) for which the series is convergent. \(\sum_{n=1}^{\infty} \frac{\ln n}{n^{p}}\) Equation Transcription: Text Transcription: p Sum over n=1 ^infty ln n/n^p

The Riemann Zeta Function The function \(\zeta\) defined by \(\zeta(s)=\sum_{n=1}^{\infty} \frac{1}{n^{s}}\) where \(s\) is a complex number, is called the Riemann zeta function. For which real numbers \(x\) is \(\zeta(x)\) defined? Equation Transcription: Text Transcription: zeta zeta(s)=sum over n=1 ^infty 1/n^s s x zeta(x)

The Riemann Zeta Function The function \(\zeta\) defined by \(\zeta(s)=\sum_{n=1}^{\infty} \frac{1}{n^{s}}\) where \(s\) is a complex number, is called the Riemann zeta function. Leonhard Euler was able to calculate the exact sum of the \(p\)-series with \(p=2\) : \(\zeta(2)=\sum_{n-1}^{\infty} \frac{1}{n^{2}}=\frac{\pi^{2}}{6}\) Use this fact to find the sum of each series. (a) \(\sum_{n=2}^{\infty} \frac{1}{n^{2}}\) (b) \(\sum_{n=3}^{\infty} \frac{1}{(n+1)^{2}}\) (c) \(\sum_{n=1}^{\infty} \frac{1}{(2 n)^{2}}\) Equation Transcription: series Text Transcription: zeta zeta(s)=sum over n=1 ^infty 1/n^s s p-series p=2 zeta(2)=sum over n=1 ^infty 1/n^2=pi^2/6 Sum over n=2 ^infty 1/n^2 Sum over n=3 ^infty 1/(n+1)^2 Sum over n=1 ^infty 1/(2n)^2

The Riemann Zeta Function The function \(\zeta\) defined by \(\zeta(s)=\sum_{n=1}^{\infty} \frac{1}{n^{s}}\) where \(s\) is a complex number, is called the Riemann zeta function. Euler also found the sum of the \(p\)-series with \(p=4\) : \(\zeta(4)=\sum_{n-1}^{\infty} \frac{1}{n^{4}}=\frac{\pi^{4}}{90}\) Use Euler’s result to find the sum of the series. (a) \(\sum_{n=1}^{\infty}\left(\frac{3}{n}\right)^{4}\) (b) \(\sum_{k=5}^{\infty} \frac{1}{(k-2)^{4}}\) Equation Transcription: series Text Transcription: zeta zeta(s)=sum over n=1 ^infty 1/n^s s p-series p=4 zeta(4)=sum over n=1 ^infty 1/n^4=pi^4/90 Sum over n=1 ^infty (3/n)^4 Sum over k=5 ^infty 1/(k-2)^4

(a) Find the partial sum \(s_{10}\) of the series \(\sum_{n-1}^{\infty} 1 / n^{4}\). Estimate the error in using \(s_{10}\) as an approximation to the sum of the series. (b) Use (3) with \(n=10\) to give an improved estimate of the sum. (c) Compare your estimate in part (b) with the exact value given in Exercise 37. (d) Find a value of \(n\) so that \(s_{n}\) is within 0.00001 of the sum. Equation Transcription: Text Transcription: s_10 Sum over n=1 ^infty 1/n^4 s_10 n=10 s_n

(a) Use the sum of the first 10 terms to estimate the sum of the series \(\sum_{n-1}^{\infty} 1 / n^{2}\). How good is this estimate? (b) Improve this estimate using (3) with \(n=10\). (c) Compare your estimate in part (b) with the exact value given in Exercise 36. (d) Find a value of \(n\) that will ensure that the error in the approximation \(s \approx s_{n}\) is less than 0.001. Equation Transcription: Text Transcription: Sum over n=1 ^infty 1/n^2 n=10 n s approx s_n

Find the sum of the series \(\sum_{n-1}^{\infty} n e^{-2 n}\)correct to four decimal places. Equation Transcription: Text Transcription: Sum_n=1 ^infty ne^-2n

Estimate \(\sum_{n-1}^{\infty}(2 n+1)^{-6}\) correct to five decimal places. Equation Transcription: Text Transcription: Sum_n=1 ^infty (2n+1)^-6

How many terms of the series \(\sum_{n-2}^{\infty} 1 /\left[n(\ln n)^{2}\right]\) would you need to add to find its sum to within 0.01? Equation Transcription: Text Transcription: sum_n=2 ^infty 1/[n(ln n)^2]

Show that if we want to approximate the sum of the series \(\sum_{n-1}^{\infty} n^{-1.001}\) so that the error is less than 5 in the ninth decimal place, then we need to add more than \(10^{11,301}\) terms! Equation Transcription: Text Transcription: sum_n=1 ^infty n^-1.001 10^11,301

(a) Show that the series \(\sum_{n-1}^{\infty}(\ln n)^{2} / n^{2}\) is convergent. (b) Find an upper bound for the error in the approximation \(s \approx s_{n}\) (c) What is the smallest value of \(n\) such that this upper bound is less than 0.05? (d) Find \(s_{n}\) for this value of \(n\). Equation Transcription: Text Transcription: sum_ n=1 ^infty (ln n)^2/n^2 s approx s_n s_n n

(a) Use (4) to show that if \(s_{n}\) is the \(n\)th partial sum of the harmonic series, then \(s_{n} \leqslant 1+\ln n\) (b) The harmonic series diverges, but very slowly. Use part (a) to show that the sum of the first million terms is less than 15 and the sum of the first billion terms is less than 22 . Equation Transcription: th Text Transcription: s_n nth S_n leq 1+ln n

Use the following steps to show that the sequence, \(t_{n}=1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}-\ln n\) has a limit. (The value of the limit is denoted by \(\gamma\) and is called Euler's constant.) (a) Draw a picture like Figure 6 with \(f(x)=1 / x\) and interpret \(t_{n}\) as an area [or use (5)] to show that \(t_{n}>0\) for all \(n\). (b) Interpret \(t_{n}-t_{n+1}=[\ln (n+1)-\ln n]-\frac{1}{n+1}\) as a difference of areas to show that \(t_{n}-t_{n+1}>0\). Therefore \(\left\{t_{n}\right\}\) is a decreasing sequence. (c) Use the Monotonic Sequence Theorem to show that \(\left\{t_{n}\right\}\) is convergent. ________________ Equation Transcription: ________________ Text Transcription: t_{n} = 1 + frac{1}{2} + frac{1}{3} + cdots + frac{1}{n} - ln n gamma f(x) = 1/x t_{n} t_{n}>0 n t_{n} - t_{n + 1} = [ln (n+1) - ln n] - frac{1}{n+1} t_{n} - t_{n+1} > 0 {t_n} {t_n}

Find all positive values of \(b\) for which the series \(\sum_{n-1}^{\infty} b^{\ln n}\) converges ________________ Equation Transcription: Text Transcription: b sum_{n-1}^{infty} b^ln n

Find all values of \(c\) for which the following series converges. \(\sum_{n=1}^{\infty}\left(\frac{c}{n}-\frac{1}{n+1}\right)\) ________________ Equation Transcription: Text Transcription: c sum_{n = 1}^{infty} (frac{c}{n} - frac{1}{n+1})